Statistical inference for generalized Ornstein-Uhlenbeck processes

TL;DR

This paper develops a new method using Mellin transforms to estimate the Lévy process characteristics in generalized Ornstein-Uhlenbeck models from low-frequency data, achieving optimal convergence rates.

Contribution

It introduces a novel Mellin transform-based estimation approach for Lévy triplet parameters in generalized Ornstein-Uhlenbeck processes, with proven optimal convergence.

Findings

Estimates attain optimal minimax convergence rates.

Algorithms are validated through numerical simulations.

Provides a new statistical inference framework for Lévy-driven processes.

Abstract

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[ X_{t} = e^{-\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u-}} d u \right), \] where \(\xi_s\) is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process \(\xi\) from the low-frequency observations of the process \(X\). We present a novel approach towards estimating the L{\'e}vy triplet of \(\xi,\) which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.